Most games have some way to make sure they cannot go on forever.
1) Chess: I had thought there was a 50-move rule and a 3-times-same-position rule, but its a byte more complicated than that, see here. There is also a chess clock. Suffice to say, Chess can never on forever (though it may seem like it does).
2) NIM: Eventually all of the stones are gone. There may be more complicated versions where you can add some stones, but in those versions I suspect that there is some parameter that goes to 0.
3) Basketball, Football, Hockey, Soccer: These all have a clock so they are time limited. For overtime there are also rules that make sure the game cannot go on forever. Or maybe its just very rare: what if the Superb Owl (spelled that way to avoid lawsuits, see here) is tied 0-0 at the end of the four quarters and goes into overtime and... nobody scores... ever. Could the game go on forever or would the referees declare it a tie? In the regular season there are ties, but in the in the superb owl? Actually this may be more a problem in the playoffs since you need to determine who goes to the next round.
4) Take your favorite game. I would bet dollars to doughnuts (what an odd phrase---see here for more about the phrase) that there is some mechanism to make sure the game ends. An exception that Darling pointed out to me: If in Gin Rummy both players are terrible then the game can go on forever. This is probably true for other games as well and actually makes the question into two questions (a) will a game terminate no matter what the players do, and (b) (not sure how to formalize) will a game terminate if both players are trying to win and are making reasonable moves.
You may have noticed that in item 3 I left out Baseball. There is no clock in baseball. So one way the game can go on forever is to have a tie and extra innings and nobody scores. I think the umpire has the authority to call it a tie. (Incidentally, the shortened baseball season has a new extra inning rule---each inning starts with a runner on second. See here,) When Lance read an earlier version of this post he pointed me to 5 ways a game can go on forever, not counting the example I have later in this post. Here is where Lance found the question and answer (look on the first page under Speed Department for the question, and the very end of the second page for the answer). I also did my own writuep with more details, see here. Also of interest (though not if you were actually at the game this happened), the record for number of times a player has a foul with 2 strikes is 16, see here.
However, I came across an example more obscure than any of those.
Here is what happened (and you can see the video of it here, though it really starts about a minute into it. Keep reading- it looks like its another post, but its part of this post: